It had previously passed the material to ex-police officer Mak Chishty, who said that 17 instances should have been reported to the police for investigation, and another four were potential race hate crimes.

LBC said the allegations included threats against MPs, including a message on Facebook that a female Labour MP was a “zionist extremist…who hates civilised people” and is “about to get a good kicking”.

‘Full force of law’

Ms Dick said she hoped to be able to conduct the investigations quickly.

“We would always want institutions and political parties and similar to be able to regulate themselves.

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“However, if somebody passes us material which they say amounts to a crime we have a duty to look at that and not just dismiss it.

“We have been assessing some material that was passed to me, in a radio studio of all things, about two months ago and we are now investigating some of that material because it appears there may have been crime committed.”

Deputy Labour leader Tom Watson said anyone connected to Labour engaging in such behaviour had no place in the party and must have the “full force of the law” applied to them.

He told Today that the police investigation would hopefully “silence” once and for all the minority who still believed that anti-Semitism did not exit within his party.

It is understood Labour has not been contacted by the police and has not been told of the exact nature of the allegations being investigated.

“The Labour Party has a robust system for investigating complaints of alleged breaches of Labour party rules by its members,” a spokesman said.

“Where someone feels they have been a victim of crime, they should report it to the police in the usual way.”

In September, Labour’s ruling body agreed to adopt in full an international definition of anti-Semitism, after a long-running row about the leadership’s handling of the issue.

It vowed to incorporate all the 11 examples of anti-Semitism cited by the International Holocaust Remembrance Alliance into its code of conduct.

But Jewish groups criticised an accompanying statement which the party said was aimed at protecting free speech.